The document discusses various number systems including decimal, binary, octal, and hexadecimal. It provides details on how to convert between these different bases including examples of converting decimal to binary, binary to decimal, hexadecimal to binary, decimal to octal, octal to decimal, hexadecimal to decimal, decimal to hexadecimal, binary to octal, binary to hexadecimal, octal to hexadecimal, and hexadecimal to octal. Conversion techniques involve grouping bits, multiplying place values, and keeping track of remainders from division.

1. Number Systems - II
ECE – B
Ist semester.

2. Common Number Systems
Used by Used in
System Base Symbols humans? computers?
Decimal 10 0, 1, … 9 Yes No
Binary 2 0, 1 No Yes
Octal 8 0, 1, … 7 No No
Hexa- 16 0, 1, … 9, No No
decimal A, B, … F

3. Hexa-
Decimal Binary Octal decimal
0 0 0 0
1 1 1 1
2 10 2 2
3 11 3 3
4 100 4 4
5 101 5 5
6 110 6 6
7 111 7 7

4. Conversion Among Bases
• The possibilities:
Decimal Octal
Binary Hexadecimal

5. Quick Example
2510 = 110012 = 318 = 1916
Base

6. Binary to Decimal
• Technique
– Multiply each bit by 2n, where n is the “weight” of
the bit
– The weight is the position of the bit, starting from
0 on the right
– Add the results

7. Example
Bit “0”
1010112 => 1 x 20 = 1
1 x 21 = 2
0 x 22 = 0
1 x 23 = 8
0 x 24 = 0
1 x 25 = 32
4310

8. Example for the fractional number.
Usha Mary Sharma. DBCET

9. Hexadecimal to Binary
• Technique
– Convert each hexadecimal digit to a 4-bit
equivalent binary representation

10. Example
10AF16 = ?2
1 0 A F
0001 0000 1010 1111
10AF16 = 00010000101011112

11. Decimal to Octal
• Technique
– Divide by 8
– Keep track of the remainder

12. Example
123410 = ?8
8 1234
154 2
8
19 2
8
2 3
8
0 2
123410 = 23228

13. Example
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14. Octal to Decimal
• Technique
– Multiply each bit by 8n, where n is the “weight” of
the bit
– The weight is the position of the bit, starting from
0 on the right
– Add the results

15. Example
7248 => 4 x 80 = 4
2 x 81 = 16
7 x 82 = 448
46810

16. Hexadecimal to Decimal
• Technique
– Multiply each bit by 16n, where n is the “weight”
of the bit
– The weight is the position of the bit, starting from
0 on the right
– Add the results

17. Example
ABC16 => C x 160 = 12 x 1 = 12
B x 161 = 11 x 16 = 176
A x 162 = 10 x 256 = 2560
274810

18. Example
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19. Decimal to Binary
• Technique
– Divide by two, keep track of the remainder
– First remainder is bit 0 (LSB, least-significant bit)
– Second remainder is bit 1
– Etc.

20. Example
12510 = ?2 2 125
2 62 1
2 31 0
15 1
2
7 1
2
2 3 1
2 1 1
0 1
12510 = 11111012

21. Example
Usha Mary Sharma. DBCET

22. Octal to Binary
• Technique
– Convert each octal digit to a 3-bit equivalent
binary representation

23. Example
7058 = ?2
7 0 5
111 000 101
7058 = 1110001012

24. Decimal to Hexadecimal
• Technique
– Divide by 16
– Keep track of the remainder

25. Example
123410 = ?16
16 1234
16 77 2
16 4 13 = D
0 4
123410 = 4D216

26. Example
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27. Binary to Octal
• Technique
– Group bits in threes, starting on right
– Convert to octal digits

28. Example
10110101112 = ?8
1 011 010 111
1 3 2 7
10110101112 = 13278

29. Binary to Hexadecimal
• Technique
– Group bits in fours, starting on right
– Convert to hexadecimal digits

30. Example
10101110112 = ?16
10 1011 1011
2 B B
10101110112 = 2BB16

31. Octal to Hexadecimal
• Technique
– Use binary as an intermediary

32. Example
10768 = ?16
1 0 7 6
001 000 111 110
2 3 E
10768 = 23E16

33. Hexadecimal to Octal
• Technique
– Use binary as an intermediary

34. Example
1F0C16 = ?8
1 F 0 C
0001 1111 0000 1100
1 7 4 1 4
1F0C16 = 174148

35. All the best.